Properties

Degree 2
Conductor $ 2^{4} \cdot 3^{3} \cdot 7 $
Sign $0.612 - 0.790i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.18i·5-s + (1 + 2.44i)7-s − 5.26i·11-s − 1.01i·13-s + 3.08i·17-s + 3.24·19-s − 0.903i·23-s + 0.242·25-s + 5.34·29-s + 5.24·31-s + (−5.34 + 2.18i)35-s + 37-s − 8.35i·41-s + 4.47i·43-s + 12.8·47-s + ⋯
L(s)  = 1  + 0.975i·5-s + (0.377 + 0.925i)7-s − 1.58i·11-s − 0.281i·13-s + 0.748i·17-s + 0.743·19-s − 0.188i·23-s + 0.0485·25-s + 0.992·29-s + 0.941·31-s + (−0.903 + 0.368i)35-s + 0.164·37-s − 1.30i·41-s + 0.682i·43-s + 1.88·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.612 - 0.790i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.612 - 0.790i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $0.612 - 0.790i$
motivic weight  =  \(1\)
character  :  $\chi_{3024} (1567, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 3024,\ (\ :1/2),\ 0.612 - 0.790i)\)
\(L(1)\)  \(\approx\)  \(2.012418122\)
\(L(\frac12)\)  \(\approx\)  \(2.012418122\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;7\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;3,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 + (-1 - 2.44i)T \)
good5 \( 1 - 2.18iT - 5T^{2} \)
11 \( 1 + 5.26iT - 11T^{2} \)
13 \( 1 + 1.01iT - 13T^{2} \)
17 \( 1 - 3.08iT - 17T^{2} \)
19 \( 1 - 3.24T + 19T^{2} \)
23 \( 1 + 0.903iT - 23T^{2} \)
29 \( 1 - 5.34T + 29T^{2} \)
31 \( 1 - 5.24T + 31T^{2} \)
37 \( 1 - T + 37T^{2} \)
41 \( 1 + 8.35iT - 41T^{2} \)
43 \( 1 - 4.47iT - 43T^{2} \)
47 \( 1 - 12.8T + 47T^{2} \)
53 \( 1 + 7.55T + 53T^{2} \)
59 \( 1 - 5.34T + 59T^{2} \)
61 \( 1 - 3.46iT - 61T^{2} \)
67 \( 1 + 3.88iT - 67T^{2} \)
71 \( 1 + 2.18iT - 71T^{2} \)
73 \( 1 - 14.2iT - 73T^{2} \)
79 \( 1 - 9.37iT - 79T^{2} \)
83 \( 1 + 12.8T + 83T^{2} \)
89 \( 1 - 3.98iT - 89T^{2} \)
97 \( 1 - 6.33iT - 97T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−8.581978989689940955989537564330, −8.299308976561922338931248660943, −7.33820438372341476327529592924, −6.41062398081760844473146472813, −5.88506579742064536606287400488, −5.17793812250280011160131129726, −3.96982527596415849892598821957, −3.02104588870076831572568379640, −2.52390058660053924227946943497, −1.03064429535488964589658588265, 0.805454613890060925437154469751, 1.69354537167868441863769487856, 2.91175835356353368399335585846, 4.23500335880941127213608801185, 4.64374306709071557329941279827, 5.26070610997266333498647939629, 6.49782564338728490722398886399, 7.24961733922511828458188219036, 7.75772655225807916400513960961, 8.624356355037225521473805711270

Graph of the $Z$-function along the critical line